**Abstract**

In this chapter, we present an extension of the rigorous analysis of the propagation of electromagnetic waves in magnetic transverse (TM) and transverse electric (TE) modes in a metallic circular waveguide partially filled with anisotropic metamaterial. In our analysis, the design of waveguide filters with uniaxial discontinuities is based on the determination of the higher-order modes, which have been analyzed and exploited. Below the cutoff frequency, the back backward waves can propagate in an anisotropic material. The numerical results with our MATLAB code for TM and TE modes were compared to theoretical predictions. Good agreements have been obtained. We analyzed a waveguide filters filled with partially anisotropic metamaterial using the mode matching (MM) technique based on the Scattering Matrix Approach (SMA), which, from the decomposition of the modal fields (TE and TM modes), are used to determine the dispersion matrix and thus the characterization of a discontinuity in waveguide. We extended the application of MM technique to the anisotropic material. By using modal analysis, our approach has considerably reduced the computation time compared to High Frequency Structure Simulator (HFSS) software.

**Keywords:** anisotropic metamaterials, forward and backward waves, MM, modal analysis, waveguides discontinuity
